The water drops that charge themselves to a spark

In 1867 William Thomson — not yet Lord Kelvin — described a device he called “a self-acting condenser”. It is one of the strangest objects in the history of electricity, because it seems to make high voltage out of nothing. A pipe of tap water, connected to earth, divides into two nozzles. Each nozzle drips through a metal ring, and each stream falls into a metal can, insulated from the bench. The left ring is wired to the right can, and the right ring to the left can. That is all.

Turn on the water and wait. For a while nothing happens. Then the streams start to behave oddly — bending, spraying — and a spark cracks across the gap between the two cans, a few millimetres of air, at around ten thousand volts. Then another, and another, every half-minute or so for as long as the water runs. Nothing was charged to begin with, nothing was rubbed against anything, and there is no battery. The charge was separated by the water falling, and the device’s own wiring did the rest. Built from two tin cans, a pair of curtain rings and a garden hose, it is still one of the cheapest ways to make ten thousand volts in a classroom, and one of the hardest to explain: most of the pieces are passive, the arrangement is symmetric, and the source of the energy is not where anyone first looks.

The charge a drop takes with it #

The whole effect rests on electrostatic induction, which the charge that has to be somewhere else followed in detail: a conductor brought near a charged object develops an opposite charge on its near side, drawn from elsewhere in itself or from the earth. A drop of water forming at the end of an earthed nozzle is a conductor connected to the earth. If the ring around it is at a positive potential, the drop draws negative charge up from the earth to its surface while it hangs there, attracted by the ring. When it breaks off, it takes that charge with it.

The amount is the drop’s capacitance to the ring times the ring’s potential. For a drop three millimetres across, that capacitance is a few tenths of a picofarad — the capacitance of an isolated sphere of that size, $4\pi\varepsilon_0 r$, which how much charge a shape will hold found growing with a body’s size — so a ring at one volt gives each drop about a sixth of a picocoulomb. Small, but every drop carries it, and many drops fall every second.

Each drop falls into its can, and the can, being a closed metal container, collects the whole of the drop’s charge: the inside of a conductor found that charge brought into a hollow conductor ends up entirely on its outside, whatever happens inside. Faraday’s ice-pail experiment showed exactly that, and the dropper’s cans are ice pails. So the left stream steadily charges the left can with charge opposite in sign to the left ring’s potential.

The crossed wires that make it run away #

Now the wiring matters. Suppose the left can starts a little positive and the right can a little negative — a stray volt from anywhere. The left ring is wired to the right can, so it is negative; the drops forming in it pick up positive charge and carry it into the left can, which becomes more positive. The right ring is wired to the left can, so it is positive; its drops pick up negative charge and carry it into the right can, which becomes more negative. Each can’s charge strengthens the other can’s ring, which strengthens the charging of the first can. The imbalance feeds itself.

The rate of charging of each can is proportional to the other can’s potential, and so, with the signs as they are, the difference between them grows in proportion to itself. That is exponential growth, at a rate

$$\gamma = \frac{n\,c}{C},$$

where $n$ is the number of drops a second in each stream, $c$ each drop’s capacitance to its ring and $C$ each can’s capacitance. With twenty drops a second, a drop capacitance of 0.17 picofarad and cans of twenty picofarads, the rate is one e-folding every six seconds. From a single volt it takes about fifty-five seconds to multiply ten thousand times, to the voltage that jumps a few millimetres of air. That matches what a demonstrator sees: a quiet minute, and then sparks.

The figures integrate the two cans’ equations step by step rather than writing down the exponential, and the measured growth agrees with $nc/C$ to better than a per cent. The physics is linear until the very end: twice the stray voltage, twice the voltage at every later moment. That linearity is what makes the starting imbalance irrelevant to how long the process takes, as long as there is some.

The symmetry and how it breaks #

The device is perfectly symmetric — swap left and right and nothing changes — and yet it always ends with one can positive and the other negative. Which one is decided by the starting imbalance, and the reason is clearest drawn as a picture of both cans at once.

The state of the device is a point in a plane: left can’s potential across, right can’s up. The dripping moves the point, and the arrows show which way at each place. They point inward along the diagonal where both cans have the same potential: if both cans are, say, positive, both rings are positive, both streams deliver negative charge, and both cans discharge. They point outward along the other diagonal, where the cans are equal and opposite: there the process runs away. The origin, with both cans uncharged, is an equilibrium — but an unstable one, a saddle, like a ball balanced on a pass between two valleys.

So whatever small, random charge the cans begin with, its common part decays and its difference grows, and the point is swept out along one arm of the diagonal or the other. Which arm depends on the sign of the starting difference: a stray volt, a charged cloud passing overhead, a static charge on the demonstrator’s jumper. The device amplifies an imbalance too small to measure into one that sparks, and its final polarity is a record of which way the initial imbalance happened to lean. It is one of the simplest physical examples of a symmetric system choosing an asymmetric state, the pattern the second law with a probability attached found behind every fluctuation that grows instead of fading.

Spark after spark #

When the voltage reaches the breakdown of the air in the gap, a spark jumps and the cans discharge. They do not quite discharge to nothing.

A spark is a brief, violent current that drains most of the cans’ charge, but it stops when the voltage across the gap falls too low to keep the ionised channel alive, leaving a residue of the same sign, and the rings and the wiring keep some charge of their own. From that residue the exponential growth starts again, so the device settles into a rhythm: a slow climb, a rush at the end, a spark, and a climb again. Because the residue keeps its sign, the cans keep the same polarity spark after spark. Discharge them completely, by touching them both with an earthed wire, and the next charging starts from whatever stray imbalance is around, and may go either way.

The breakdown voltage is set by the gap. The spark that needs room to start found that air at ordinary pressure breaks down at about three million volts per metre across gaps of a few millimetres, so a three-millimetre gap sparks near ten thousand volts. A wider gap delays the spark and makes it louder; a gap that is too wide lets the voltage reach the device’s own ceiling first.

Where the energy comes from #

A spark at ten thousand volts carries real energy, and something has to supply it. Nothing in the device is a battery. The supply is gravity.

Each drop reaching its can carries charge of the same sign as the can — that is the whole point — so as it approaches, it is repelled. It reaches the can only because it is falling, and the work it does against the can’s repulsion is paid for out of its gravitational energy. The electrical energy delivered per drop is roughly its charge times the can’s potential, and since its charge is itself proportional to that potential, the energy grows as the square of the voltage, $cV^2$. When that equals the drop’s gravitational energy over its fall, $mgh$, the drops can no longer reach the can: they slow, hover and are deflected aside, and the stream visibly bends away. That sets a ceiling,

$$V_{\max} = \sqrt{\frac{m g h}{c}},$$

about 16 kilovolts for three-millimetre drops falling thirty centimetres. In practice the spark usually comes first, but a demonstrator who widens the gap can watch the streams spray sideways as the ceiling is approached. Every spark is the gravitational energy of thousands of drops, converted to electrical energy by forcing charged drops into a can that repels them.

Drops that tear themselves apart #

Long before the gravitational ceiling, something else limits the drops. A charged drop’s charge pushes outward on its own surface, the electrostatic pressure that the pressure a charge puts on its own metal found pulling at any charged conductor, and only surface tension holds the drop together against it. Lord Rayleigh worked out in 1882 that a drop becomes unstable when its charge exceeds $8\pi\sqrt{\varepsilon_0\gamma r^3}$, with $\gamma$ the surface tension: for a water drop three millimetres across, about one nanocoulomb. A drop forming in a ring at ten kilovolts would carry more than that — 1.7 nanocoulombs by the drop-capacitance estimate — and it cannot: near that voltage the drops do not form cleanly at all, but are pulled into fine jets and break into charged spray, the same instability that the thread that cannot stay a thread found breaking a liquid column into droplets, here driven by charge rather than by surface tension alone.

That is the spraying a demonstrator sees as the sparks approach, and it is a working technology in its own right. Electrospray, in which a liquid is pushed out of a fine needle held at a few kilovolts and breaks into a mist of charged droplets, is how large biological molecules are lifted gently into the vacuum of a mass spectrometer, and the method won John Fenn a share of the 2002 Nobel Prize in chemistry. In the dropper it is a nuisance that wastes charge; in the spectrometer it is the point.

Why it needs water, and why tap water will do #

The drop must be able to fill with induced charge while it forms, which takes it a few tens of milliseconds. Charge reaches a conductor’s surface in a time set by its permittivity divided by its conductivity — the relaxation time of the material — and for tap water, which conducts reasonably well because of its dissolved salts, that is far less than a microsecond. Even distilled water, a thousand times less conductive, relaxes in a few microseconds, still thousands of times faster than a drop forms. An insulating liquid such as oil would not work: its charge could not reach the surface of the forming drop before the drop fell, and the induced charge would never be carried away. The water’s job is to be a conductor that can be cut into pieces and dropped, carrying its induced charge with each piece — which is why the device is a dropper of water and not of anything else that flows.

How long the wait is #

The time to the first spark depends on only three quantities.

The growth rate is the drip rate times the drop’s capacitance to its ring, divided by the can’s capacitance. Faster dripping charges the cans faster; larger drops, with more capacitance to their rings, carry more charge each; larger cans need more charge per volt and take longer to charge, though they then give a fatter spark. The starting voltage enters only through a logarithm, so a stray volt and a stray millivolt differ by about forty seconds: the device is insensitive to how it starts, as long as it does not start exactly balanced.

A dynamo made of water #

Kelvin understood the dropper as an electrostatic version of something just invented. In 1866 and 1867 Werner von Siemens, Charles Wheatstone and others independently built the first self-excited dynamos: generators whose field coils are powered by the generator’s own output, so that a weak residual magnetism in the iron produces a small current, which strengthens the field, which produces a larger current, and the output builds up exponentially from almost nothing. The water dropper and the self-excited dynamo are the same loop — an output fed back to strengthen its own cause — and both choose their polarity from whatever residue they start with.

The Earth’s magnetic field is a self-excited dynamo of molten iron, and the field no symmetric flow can keep found the conditions it must meet. It too has a polarity that its symmetric equations do not prefer, and it has flipped hundreds of times in the geological record, each time choosing anew. Thunderclouds separate charge by a process with the same ingredients — falling particles, induction and gravity doing the work — as the ice and graupel inside them collide and part with opposite charges and fall at different speeds; the charge they build up drives the planet’s whole electrical circuit, which the field that keeps Greenwich time followed round the globe.

Leakless cans, spherical drops and a tidy spark #

The model treats the charging as perfectly linear and the cans as perfect capacitors, with no leakage. Real devices leak: humid air, dust and the insulating supports all conduct a little, and at high voltage corona from sharp edges bleeds charge away, so in damp weather a dropper may never reach a spark at all. The drop’s capacitance is taken as that of an isolated sphere; a drop forming inside a ring, attached to a nozzle, has a somewhat different capacitance that depends on the geometry, and real drops vary in size. The figures ignore the charged drops’ effect on one another and on the streams, which matters near the ceiling, where the streams spray.

The spark model is a simple reset to a fixed residue, where real sparks leave a residue that varies from spark to spark; and the starting imbalance is taken as one volt, where in practice it is whatever happens to be there. None of these change the qualitative behaviour: exponential growth from any imbalance, a chosen polarity, a spark, and a ceiling set by gravity.

Still open: how small a starting charge matters #

The dropper amplifies the starting imbalance by a factor of ten thousand in under a minute, and in principle it would amplify any imbalance, however small — down to the thermal fluctuations of charge on the cans themselves, the noise half a kT in a piece of wire found in any conductor. In practice the starting imbalance is always set by something much larger: stray fields, contact potentials between the water and the metal of the nozzle, charges on nearby clothing. Whether a carefully shielded, perfectly symmetric dropper would be started by thermal noise alone, and how the statistics of its first polarity would then look, has not been measured; the instrument’s own asymmetries are hard to make smaller than the effect.

What the dropper shows is a general pattern in its plainest form. When a system’s output feeds back to strengthen its own cause, any small imbalance grows exponentially, and a perfectly symmetric device ends in an asymmetric state chosen by the imbalance it began with; the energy for the growth must come from somewhere, and here it comes from gravity, one falling drop at a time. Two streams of tap water and a pair of crossed wires are enough to turn a stray volt into a spark.